For [tex]f(x)=x+2[/tex] and [tex]g(x)=2 x+3[/tex], find the following.
a. [tex](f \circ g)(x)[/tex]
b. [tex](g \circ f)(x)[/tex]
c. [tex](f \circ g)(5)[/tex]
a. [tex](f \circ g)(x)=[/tex] (Simplify your answer.)
Answer (1)
Find ( f ∘ g ) ( x ) by substituting g ( x ) into f ( x ) : ( f ∘ g ) ( x ) = f ( g ( x )) = f ( 2 x + 3 ) = 2 x + 3 + 2 = 2 x + 5 .
Find ( g ∘ f ) ( x ) by substituting f ( x ) into g ( x ) : ( g ∘ f ) ( x ) = g ( f ( x )) = g ( x + 2 ) = 2 ( x + 2 ) + 3 = 2 x + 4 + 3 = 2 x + 7 .
Find ( f ∘ g ) ( 5 ) by substituting x = 5 into ( f ∘ g ) ( x ) : ( f ∘ g ) ( 5 ) = 2 ( 5 ) + 5 = 10 + 5 = 15 .
The final answers are ( f ∘ g ) ( x ) = 2 x + 5 , ( g ∘ f ) ( x ) = 2 x + 7 , and ( f ∘ g ) ( 5 ) = 15 . Therefore, ( f ∘ g ) ( x ) = 2 x + 5 .
Explanation
Understanding the Problem We are given two functions, f ( x ) = x + 2 and g ( x ) = 2 x + 3 . We need to find the composite functions ( f ∘ g ) ( x ) and ( g ∘ f ) ( x ) , and then evaluate ( f ∘ g ) ( 5 ) .
Finding (f \circ g)(x) To find ( f ∘ g ) ( x ) , we need to substitute g ( x ) into f ( x ) . This means we replace x in f ( x ) with g ( x ) , which is 2 x + 3 . So, we have f ( g ( x )) = f ( 2 x + 3 ) = ( 2 x + 3 ) + 2. Simplifying this expression, we get f ( g ( x )) = 2 x + 3 + 2 = 2 x + 5. Thus, ( f ∘ g ) ( x ) = 2 x + 5 .
Finding (g \circ f)(x) To find ( g ∘ f ) ( x ) , we need to substitute f ( x ) into g ( x ) . This means we replace x in g ( x ) with f ( x ) , which is x + 2 . So, we have g ( f ( x )) = g ( x + 2 ) = 2 ( x + 2 ) + 3. Simplifying this expression, we get g ( f ( x )) = 2 x + 4 + 3 = 2 x + 7. Thus, ( g ∘ f ) ( x ) = 2 x + 7 .
Finding (f \circ g)(5) To find ( f ∘ g ) ( 5 ) , we substitute x = 5 into the expression we found for ( f ∘ g ) ( x ) , which is 2 x + 5 . So, we have ( f ∘ g ) ( 5 ) = 2 ( 5 ) + 5 = 10 + 5 = 15. Thus, ( f ∘ g ) ( 5 ) = 15 .
Final Answer Therefore, we have: a. ( f ∘ g ) ( x ) = 2 x + 5 b. ( g ∘ f ) ( x ) = 2 x + 7 c. $(f \circ g)(5) = 15
Examples
Composite functions are useful in many real-world scenarios. For example, consider a store that is offering a discount on all of its items. Let f ( x ) be the price of an item after a 20% discount, and let g ( x ) be the price of an item after a $5 coupon is applied. Then ( f ∘ g ) ( x ) represents the price of an item after the $5 coupon is applied first, followed by the 20% discount. Understanding composite functions allows you to model and analyze such situations effectively.
